Remarks on stability and diffusion in high-dimensional Hamiltonian systems and partial differential equations

Ergodic Theory and Dynamical Systems

Volumn 24, Issue 5, 2004, Pages 1331-1357

  Bourgain, Jean ^a  

^a INSTITUTE FOR ADVANCED STUDY   (United States)

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EID: 9344246847     PISSN: 01433857     EISSN: None     Source Type: Journal    
DOI: 10.1017/S0143385703000750     Document Type: Article

References (23)

1. [Bam] D. Bambusi. Nekhoroshev theorem for small amplitude solutions in nonlinear Schrödinger equations. Math. Z. 230 (1999), 345-387.
2. [B1] B. Bourgain. Nonlinear Schrödinger Equations (Park City Lectures). American Mathematical Society, Providence, RI 1999.
3. [B2] J. Bourgain. Growth of Sobolev norms in linear Schrödinger equations with quasi-periodic potential. Comm. Math. Phys. 204(1) (1999), 207-257.
4. [B3] J. Bourgain. On the growth of Sobolev norms in linear Schrödinger equations with smooth time dependent potential. J. Math. 77 (1999), 315-348.
5. [B4] J. Bourgain. On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE. Internat. Math. Res. Notices 6 (1996), 277-304.
6. [B5] J. Bourgain. Greens function estimates for lattice Schrödinger operators and applications. Ann. Math. Studies to appear.
7. [B6] J. Bourgain. Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations. Ann. Math. 148 (1998), 463-439.
8. [B7] J. Bourgain. Construction of approximative and almost periodic solutions of perturbed linear Schrödinger and wave equations. Geom. Funct. Anal. 6 (1996), 201-230.
9. [B8] J. Bourgain. On diffusion in high-dimensional Hamiltonian Systems and PDE. J. Anal. Math. 80 (2000), 1-35.
10. [BFG] G. Benettin, J. Fröhlich and A. Giorgili. A Nekhoroshev-type theorem for Hamiltonian systems with infinitely many degrees of freedom. Comm. Math. Phys. 119 (1989), 95-108.
11. [Cr] W. Craig. Problèmes de petits diviseurs dans les équations aux dérivées partielles. Panoramas et Synthèses. Société Mathématique de France, Paris, 2000.
12. [DJLS] R. del Rio, S. Jitomirskaya, Y. Last and B. Simon. Operators with singular continuous spectrum, IV. Hausdorff dimensions, rank one perturbations, and localization. J. Anal. Math. 69 (1996), 153-200.
13. [FSW] J. Fröhlich, T. Spencer and E. Wayne. Localization in disordered, nonlinear dynamical systems J. Stat. Phys. 42 (1986), 257-275.
14. [K] S. Kuksin. Analysis of Hamiltonian PDE (Oxford Lecture Series in Mathematics and Applications, 19). Oxford University Press, Oxford, 2000.
15. [KP] S. Kuksin and J. Pöschel. Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation. Ann. Math. (2) 143 (1996), 149-179.
16. [Lo] P. Lochak. Canonical perturbation theory via simultaneous approximation. Usp. Mat. Nauk 47(6) (1992), 59-140.
17. [N] N. Nekhoroshev. An exponential estimate of the time of stability of nearly integrable Hamiltonian systems. Usp. Mat. Nauk 32(1) (1977), 5-66.
18. [P] J. Pöschel. On the construction of almost periodic solutions for a nonlinear Schrödinger equation. Ergod. Th. & Dynam. Sys. 22 (2002), 1537-1549.
19. [PM] R. Perez-Marco. KAM Methods in PDE. Sem. Bourbaki, Paris, June 2002.
20. [PT] J. Pöschel and E. Trubowitz. Inverse Spectral Theory. Academic Press, Boston, MA, 1987.
21. [St] G. Staffilani. Quadratic forms for a 2D semilinear Schrödinger equation. Duke Math. J. 86 (1997), 79-108.
22. [W] C. Wayne. On the elimination of nonresonant harmonics. Comm. Math. Phys. 103 (1986), 351-386.
23. [ZS] V. Zakharov and A. B. Shabat. Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys.-JETP 34 (1972), 62-69.